What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros?

More precisely, suppose that $f_i: R^n \to R^m$ is a sequence of smooth functions converging uniformly in all derivatives to a function $f: R^n \to R^m$. Suppose that $z_{i,k} \in R^n$ are distinct zeroes of $f_i$ for $k = 1,\ldots, l$ converging to $0$ as $i \to \infty$ for all $k$. Clearly some partial derivatives of $f$ have to vanish at $0$, but which ones depends on how the points $z_{i,k}$ converge to $0$.

Let $d(l,n,m)$ be the function such that for any such situation above for $l,n,m$ fixed, there exists a direction $v \in R^n$ for which the first $d(l,n,m)$ directional derivatives of $f$ in the direction of $v$ vanishes at $0$. What is $d(l,n,m)$?

This is related to the question of which partial derivatives at $0$ are approximated by finite difference operators constructed from the points $z_{i,k}$. My question is how this depends on the limiting geometry of $z_{i,k}$, and what statements are independent of the geometry. There must be a literature on this, but I am having trouble finding the right "tag".